Twin Circles of Archimedes

The arbelos (ancient Greek for "shoemaker's knife") is a geometric figure that was already explored by Archimedes from Syracuse. It is bounded by three semicircles and is shown here in green color. 

Click on the image.

An interesting property becomes visible if the common tangent of the two small semicircles is added (vertical line). In the two parts of the arbelos that are separated by this line, we draw the largest circles (violet) that still fit. Because these circles have the same radius, they are called the twin circles of Archimedes. 

Drag the mouse with pressed mouse button to modify the vertical line and the two smaller semicircles. If you check one or both of the checkboxes, some auxiliary lines will be drawn on the left and/or right. These lines are needed to calculate the radii of the circles using the Pythagorean theorem and thus prove their equality.